calibration of partial safety factors ... - standard.no anlegg og eiendom/2015 eurokoder/6... ·...

11.12.2015

Calibration of Partial Safety Factors

1

Calibration of Partial Safety

Factors

Jochen KöhlerNTNU

11.12.2015


2

Outline and scope The generic design equation.

Calibration of design equations with one variable load.

Calibration of design equations with two variable loads.

Notional reliability and hidden safety.

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Representation of a generic design situation

( )( )* * ** 1 m

G k Q kk

z G QRγ α γ α γ= + −( )( )

** * 1 k

G k Q km

z R G Qα γ α γγ

≥ + −

kG k Q k

m

z R G Qγ γγ

≥ +

( ) ( )* * * * 1 0g z R G Qξ α α= − − − ≤X

Design equation:

Different relations of 𝐺𝐺 and 𝑄𝑄

Generalization with normalized variables and G

G Qα =

+

Estimation of failure probability based on the limit state function:

( ) ( )( )Pr Pr 0F g= ≤X

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Representation of a generic design situation Example Steel structure

Variable Distribution COV p.s.f.

yield strength lognormal 0.07 1.05

MU_steel lognormal 0.05 -

dead load normal 0.1 1.35

snow load Gumbel 0.6 1.5

( )( )* * ** 1 m

G k Q kk

z G QRγ α γ α γ= + −

( ) ( )* * * * 1 0g z R G Qξ α α= − − − ≤X

( ) ( )( )Pr Pr 0F g= ≤X

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6

Steel - NOT Calibrated

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6

Steel - NOT Calibrated - target

ij -

target







( )2target i iMin w β β−∑

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6


Steel - Calibrated - target

ij -

target


Variable Distribution COV p.s.f. p.s.f. (cal., w=1)

yield strength lognormal 0.07 1.05 1.35

MU_steel lognormal 0.05 - -

dead load normal 0.1 1.35 1.11

snow load Gumbel 0.6 1.5 1.76

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6


Steel - Calibrated - target

ij -

target


Variable Distribution COV p.s.f. p.s.f. (cal., w=1) p.s.f. (cal., w=sp)

yield strength lognormal 0.07 1.05 1.35 1.06

MU_steel lognormal 0.05 - - -

dead load normal 0.1 1.35 1.11 0.97

snow load Gumbel 0.6 1.5 1.76 2.44

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6


Steel - Calibrated

- target

ij -

target

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Representation of a generic design situation Example Steel, Concrete, Timber structure






concrete cap. lognormal 0.15 1.5

MU_concrete lognormal 0.1 -

timber cap. lognormal 0.3 1.3

MU_timber lognormal 0.1 -

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6


Concrete - NOT Calibrated

Timber - NOT Calibrated

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6

Steel - Calibrated

Concrete - Calibrated

Timber - Calibrated

target


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Variable Distribution COV p.s.f. p.s.f. calibr.





concrete cap. lognormal 0.15 1.5 1.33

MU_concrete lognormal 0.1 - -

timber cap. lognormal 0.3 1.3 1.63

MU_timber lognormal 0.1 - -

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Representation of a generic design situation Two variable loads

(Eq. 6.10)

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Representation of a generic design situation Example Steel, Concrete, Timber structure, two var. loads






wind load Gumbel 0.6 1.5

concrete cap. lognormal 0.15 1.5

MU_concrete lognormal 0.1 -

timber cap. lognormal 0.3 1.3

MU_timber lognormal 0.1 -

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1


Q

0.500

0.5

G

2

4

6

1

Concrete - NOT Calibrated

1

Q

0.500

0.5

G

2

4

6

11

Timber - NOT Calibrated

Q

0.500

0.5

G

2

4

6

1

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1

Q

Steel - Calibrated

0.500

0.5

G

2

4

6

11

Concrete - Calibrated

Q

0.500

0.5

G

6

2

4

11

Timber - Calibrated

Q

0.500

0.5

G

2

4

6

1


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Variable Distribution COV p.s.f. p.s.f. calibr.





wind load Gumbel 0.6 1.5 1.49





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(Eq. 6.10a,b)

{ }

( ) ( )( )( ) ( )( )( )

3 4 5

3 01 1 02 2

4 1 02 2 1

5

max , ,

1 Q 1 (Eq. 6.10a)

1 1 Q (Eq. 6.10b with Q leading)

1

mG G k G Q Q k Q Q k

k

mG G k G Q Q k Q Q k

k

mG G k G Q

k

z z z z

z G QR

z G QR

z GR

γ α γ α α γ ψ α γ ψ

γ α ξ γ α α γ α γ ψ

γ α ξ γ α α γ

=

= + − + −

= + − + −

= + − ( )( )01 1 2 2 Q 1 (Eq. 6.10b with Q leading)Q k Q Q kQψ α γ + −

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G

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6Steel - Comparison Eq. 6.10a&b with &6.10

Eq. 6.10a

Eq. 6.10b

Eq. 6.10

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Variable Distribution COV p.s.f. p.s.f. calibr.(Eq. 6.10a&b)





Wind load Gumbel 0.6 1.5 1.67





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Summary and Discussion Partial safety factors can be calibrated in order to reach

target reliability “as best as possible” for generalized design equations.

The results are conditional on the assumptions. Caution is necessary, since engineering models contain a

multitude of conservative assumptions.

All considerations are only valid for linear design equations.

calibration of partial safety factors ... - standard.no anlegg og eiendom/2015 eurokoder/6... ·...

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